## 1. Introduction

Water vapor supersaturation is a key thermodynamic parameter in the formation and development of warm and cold clouds. The activation of cloud condensation nuclei (CCN), the spectrum of droplet sizes, and therefore, precipitation heavily depend on the supersaturation budget of an evolving cloud. By extension, the study of supersaturation is fundamental to determine the radiative properties of cloud fields which constitute a major climate feedback (Khain and Pinsky 2018), yet a major source of uncertainty in climate projections (Zelinka et al. 2020).

Due to its microphysical character, supersaturation cannot be explicitly resolved by global circulation models (GCMs), and hence, it has to be prescribed using adequate parameterization schemes. One way of doing this is by predicting the probability of encountering a cloud parcel at a certain relative humidity level in terms of the background dynamics and large-scale information. Taking the GCM perspective, the idea is to infer the subgrid cloud properties out of the prognosed variables: vertical velocity, temperature, specific humidity, and pressure, which altogether depict the relative humidity configuration of a homogeneous cloud parcel. However, localized temperature gradients and turbulence at the smaller scales can create inhomogeneous fluctuations which alter the spread of supersaturation values, hence changing the microphysical properties of the subgrid cloud parcel.

Turbulence, in fact, influences the life cycle of clouds, from the smallest warm cumuli to the coldest stratiform clouds (Shaw 2003). Indeed, the incorporation of turbulent models into the determination of microphysical properties has played a key role in explaining the broadening of droplet size spectra in developing cumulus clouds (Grabowski and Abade 2017; Abade et al. 2018) and mixed-phase clouds (Hoffmann 2020; Chen et al. 2023), as well as in the activation and maintenance of long-lasting supercooled liquid water in icy clouds (Korolev and Field 2008; Hoffmann 2020). Moreover, laboratory experiments suggest that turbulence at the smallest scales are also responsible for CCN activation even in mean-subsaturated environments (Prabhakaran et al. 2020).

It is crucial, then, to determine not only the mean value of supersaturation at a given location, but also its variance or even higher moments. To this end, the equations for supersaturation or condensational growth are coupled with a suitable stochastic forcing law that captures the effects of turbulence affecting both vertical velocity fluctuations (Bartlett and Jonas 1972; Sardina et al. 2015) as well as the integral radius of the droplet distribution (Manton 1979; Cooper 1989). Hence, the classical deterministic models become stochastic differential equations (SDEs), which have been widely used in the cloud physics community; see, e.g., McGraw and Liu (2006), Paoli and Shariff (2009), Sardina et al. (2015), and Siebert and Shaw (2017). In fact, this stochastic physics framework has been employed in the elaboration of analytically tractable parameterization schemes for mixed-phase clouds (Furtado et al. 2016), the study of droplet growth by condensation (Bartlett and Jonas 1972; Sardina et al. 2015), and the determination of steady-state warm cloud properties (Siewert et al. 2017).

The supersaturation equation, also known as Squires equation (Squires 1952), consists of a nonlinear relation between supersaturation with its sources and sinks, and has extensively been studied; see, e.g., Korolev and Mazin (2003) or Devenish et al. (2016)—we also refer to Eq. (2) of this paper. The addition of turbulent updrafts modeled by stochastic processes into the linearized Squires equation allowed to obtain Gaussian formulas for the ice-supersaturation (probability) distribution (Field et al. 2014), which serves to diagnose the amount of supercooled liquid water in mixed-phase clouds (Furtado et al. 2016). Along these lines, Sardina et al. (2015) also provided a relation for the moments of supersaturation and droplet size distribution. However, some restrictions were imposed in these works. First, the full, nonlinear form of Squires equation has never been examined in the stochastic context, unlike its deterministic counterpart [Korolev and Mazin 2003, Eqs. (9) and (10)]. In fact, this nonlinear form is specially relevant for clouds experiencing high maximum supersaturation, as pointed out in Devenish et al. (2016). Second, in the limit of fast turbulent decorrelation—relative to supersaturation time scales—the stochastic turbulence becomes white noise, which can be a strong assumption if one is concerned with other shorter relevant time scales like that at the CCN activation level, as already warned by Field et al. (2014) and Abade et al. (2018). Third, the time invariance of mean droplet radius—the quasi-steady assumption (Squires 1952)—appears critical to obtain analytical supersaturation distributions (Field et al. 2014; Furtado et al. 2016). Turbulence, mixing, and entrainment, however, can provoke fluctuations in droplet integral radius (Manton 1979; Cooper 1989), which consequently perturb the supersaturation budget. This work is concerned with analytically assessing and lifting the listed assumptions in the study of the supersaturation equation.

This paper is structured as follows. In section 2, the Squires equation for the evolution of supersaturation is derived, and second, a stochastic equation for turbulent updrafts is presented, in the lines of the theory of stochastic Lagrangian turbulent models; see, e.g., the work of Rodean (1996). In section 3, the quasi-steady equation—which assumes a constant mean droplet/particle radius—is analyzed in a number of approximations providing new formulas for the probability distribution of supersaturation. In section 4, fluctuations in droplet size are allowed, and their net effects on supersaturation distribution are investigated in analytical terms. A total of five different supersaturation distributions are derived. In section 5, the relevance of the five obtained probability density functions is discussed and compared in the context of mixed-phase clouds: we compare the supercooled liquid cloud fraction and liquid water contents predicted by each distribution. Finally, a discussion over the results is done in section 6. To supplement the information in the main text, appendixes are included to discuss some technical topics related to the analysis of stochastic differential equations.

## 2. The supersaturation equation

### a. The Squires equation

We consider a vertically moving cloud parcel containing a monodisperse family of liquid water droplets or ice particles—allowing mixed-phase conditions—that are spatially uniformly distributed. Furthermore, it is assumed that the number of droplets/particles per unit mass is constant in time. The evolution of supersaturation in the cloud is, essentially, determined by the sources and sinks of relative humidity due to the adiabatic cooling of the parcel in ascent, and condensation of water vapor onto the existing droplets’ or particles’ surface (Rogers and Yau 1989). However, the exact relation between the rate of change of supersaturation and its sinks and sources is obtained by taking its derivative with respect to time.

*e*is the water vapor pressure and

*E*is the same, although at saturation over a flat surface of liquid water or ice. We shall not specify now whether we are dealing with water or ice supersaturation because the stochastic analysis will be done independently. It is noted, however, that the calculations immediately below can be done for single or mixed-phase clouds; see Korolev and Mazin (2003).

*S*and employing the mass conservation, the Clausius–Clapeyron equation, and the quasi-hydrostatic approximation, Squires (1952) derived an equation to describe the evolution of the supersaturation budget—see also the appendix in the paper of Korolev and Mazin (2003):

*w*and

*q*are the vertical velocity and liquid-water or ice mixing ratio, respectively. The parameters

*a*and

*b*are the adiabatic and microphysical constant: see appendix A for the precise definition. The evolution of

*S*obeys an equation with a nonlinear term—

*b*(1 +

*S*)

*dq*/

*dt*—stemming from the time and temperature dependence of the equilibrium water vapor pressure

*E*in Eq. (1). Equation (2) reveals that only to leading order in small values of

*S*do we obtain a linear dependence on vertical velocity

*w*and water vapor condensation/deposition,

*dq*/

*dt*. Such is the case of warm clouds, where supersaturation levels do not typically exceed a few percent (Prabha et al. 2011). However, mixed-phase conditions in stratiform clouds arise precisely when ice supersaturation fluctuates spreading beyond small values (Korolev and Field 2008; Field et al. 2014), making the nonlinearity in Eq. (2) more relevant. This will be discussed in the next section.

*dq*/

*dt*. The latter is described as [Korolev and Mazin 2003, Eq. (5)]

*h*is the normalized particle size distribution and the different variables are understood for either liquid droplets or ice particles. The parameters

*N*and

*ρ*denote the particle concentration and density of air, respectively. It is worth noting that the first integral ranging from zero to unity refers to the possible distribution of capacitances

_{a}*c*, which reflects the efficiency with which droplets/ice particles collect water vapor (Rogers and Yau 1989). While the capacitance of droplets is approximately 1—since all of them are almost spheres—ice particles possess a wider range of possible shapes that are displayed under certain temperature and humidity conditions, affecting the collection of water vapor. Also, a degree of freedom is left for the density of each particle,

*ρ*, which again, for the case of ice phase, it can range between different values from particle to particle.

*A*

_{∘}is defined in the appendix A. Since we are considering a monodisperse particle size and shape, the previous equation does not only model the diffusional growth of a single particle but of the whole population. Inserting Eq. (4) into Eq. (3) we find

*S*, the expression for water vapor absorption in Eq. (5) is substituted into Eq. (2). The updraft term is not yet specified, although we anticipate that an SDE for updrafts will be coupled to the supersaturation equation in section 2b.

### b. Stochastic model for updrafts

*S*, which is allowed to be negative in case of dry, subsaturated air. The vertical velocity

_{E}*w*in such cloud parcel is assumed to be decomposed into its mean

*w*′ so that

*w*′ is related to the turbulent kinetic energy (TKE) of the background flow in the following way:

*ε*provide an estimate of the exponential rate, 1/

*τ*, at which the fluctuations

_{d}*w*′ decorrelate in time. Such a number is called the Lagrangian decorrelation time scale (Rodean 1996). Roughly speaking

*τ*> 0 indicates the amount of time needed for a turbulent flow to forget its original configuration:

_{d}*C*

_{0}is the Lagrangian structure-function constant and

*ε*is the eddy dissipation rate. Hence, if we assume that vertical motion is homogeneous, random, with stationary mean

*τ*, the simplest model for the vertical velocity is the following red noise equation (Rodean 1996, section 3.5):

_{d}*W*denotes a standard Wiener process, which accounts for the acceleration increments over

_{t}*dt*time units, that come from random pressure fluctuations which decorrelate instantly on time. Equation (9) has the structure of red noise or an Ornstein–Uhlenbeck process, for which there exists a vast collection of analytical results—see, e.g., Uhlenbeck and Ornstein (1930) or Pavliotis (2014). In particular, this one-dimensional Gaussian process satisfies the following mean, variance, and correlation properties:

*w*(0) is the initial value of the vertical velocity. Thus, as

*t*tends to infinity, the solution of Eq. (9) will distribute according to a Gaussian function with mean

*S*. In other words, turbulent mixing rates determine the characteristic time

_{E}*τ*

_{mix}to homogenize a cloud volume with its surrounding reservoir (Baker et al. 1984; Khain and Pinsky 2018):

*L*is the characteristic length of the turbulent zone. Therefore, a more general form for the supersaturation equation is studied so that mixing at the cloud edges is also taken into account:

## 3. Quasi-steady model statistics

*r*

^{2}(0), it is expected that it will remain almost constant, at least for the interval where the following inequality is satisfied; see also Korolev and Mazin (2003) and Devenish et al. (2016):

*r*in Eq. (3) to be constant, equal to

*r*and reads as

*r*is no longer a variable and that is absorbed into the constant

*B*and

*B*

_{0}is defined in the appendix A. The parameter

*B*

^{−1}is therefore the time scale associated with microphysical processes. The mixing time-scale parameter

*C*

^{−1}is here treated as independent with respect to TKE. We are therefore aiming at analyzing supersaturation statistics as a function of the different time scales involved in Eq. (14), so that the characteristic length scales

*L*are susceptible of changing if different values of TKE are considered. The target of this section is to derive analytically the stationary statistics of model (14) in a variety of approximations that are detailed in the subsections bellow.

### a. Fast Lagrangian decorrelation time scale

*B*+

*C*, as the characteristic time for the consumption of supersaturation, and the other is the Lagrangian decorrelation time-scale

*τ*. When the Lagrangian decorrelation time scale is small, the turbulent flow takes less time to forget its initial configuration compared to the typical time of approach to the equilibrium of supersaturation. This is, in the limit of (

_{d}*B*+

*C*)

*τ*→ 0,

_{d}*w*(

*t*) will become a stochastic delta-correlated process relative to the evolution of supersaturation; see also Sardina et al. (2015). Indeed, by referring to the theory of homogenization (Pavliotis and Stuart 2008, chapter 11, result 11.1), we are able to, mathematically rigorously, reduce the two-dimensional system describing

*S*and

*w*to

*A*has been introduced and defined as

*S*multiplies the Wiener increment in Eq. (16):

*A*(1 +

*S*)

*dW*—we have to specify the stochastic calculus formalism being employed. For simplicity, Eq. (16) shall be studied under the Itô formalism, although the Stratonovich viewpoint can be taken instead (Gardiner 2009). For completeness, we note that in order to convert the Stratonovich version of Eq. (16) into Itô, we apply the Itô-to-Stratonovich correction

_{t}*H*(

*S*) which reads as (Gardiner 2009; Pavliotis 2014)

*S*is now modified by the term −

_{E}*A*

^{2}, resulting from the mere presence of multiplicative noise. The best choice of stochastic formalism is not discussed here, although this equation reveals that the full form of the Squires equation encodes nonlinear interactions between supersaturation and turbulent fluctuations which yield nontrivial corrections to the stochastic formulation of supersaturation evolution.

*f*(

*S*,

*t*) indicates—when normalized—the probability of encountering a supersaturation of

*S*at time

*t*. Roughly speaking, this equation says that a density function is advected and diffused by the linear and stochastic components of Eq. (16), respectively.

*f*cannot be Gaussian. Indeed, after successive integrations detailed in appendix B, the normalized time-independent density is

*α*,

*β*, and

*α*is the nondimensional ratio of the turbulent and mixing time scales. Hence, it follows that a large value of

*α*≫ 1 yields a smaller supersaturation variance. Moreover, in the limit of large

*α*, the function

*f*

_{1}can be recast into a Gaussian by means of Laplace’s method; see, e.g., Butler (2007).

*t*tends to infinity, the expected value of

*S*will converge to an equilibrium value at a characteristic rate given by the phase relaxation

*τ*(Korolev and Mazin 2003). If the dynamics were deterministic, such rate would be given by

_{p}^{3}noise realizations and averaging them at every time step. The procedure is repeated 10 times to obtain the mean (blue line) and standard deviation (gray shade). We also include the relaxation curve for the linear approximation that will be explained in the next section. The initial condition of supersaturation is taken to be 0.5 for demonstration purposes. It is observed that the deterministic relaxation (orange curve) provides a good estimate for the stochastic one (blue curve).

#### Linear approximation

*S*≈ 1 is widely taken in the literature as a first-order approximation of the Squires equation (Pruppacher and Klett 2010; Khain and Pinsky 2018). Under this approximation and some algebraic manipulations, Eq. (16) becomes an OU process—which was already mentioned earlier around Eq. (9):

*t*tends to infinity, the supersaturation value subject to a constant mean updraft tend to an equilibrium value

*B*+

*C*). Hence, the steady-state statistics are provided by a Gaussian distribution:

### b. Slow Lagrangian decorrelation time scale

*τ*/

_{d}*τ*going to zero cannot be taken and therefore the white noise approximation of the previous section is not valid, as already pointed out in Field et al. (2014) and Abade et al. (2018). For instance, if number concentration increases—thus reducing

_{p}*τ*—the white noise approximation does not hold any more. Here we provide a formula for the distribution of supersaturation in the presence of turbulent updraft fluctuations with exponentially decaying correlation rate

_{p}*S*≈ 1. We obtain the following two-dimensional stochastic linear equation:

**W**

*is a two-dimensional independent Wiener process and where the matrices*

_{t}**m**are defined as

*S*we apply the affine projection

*P*= [1, 0] to the random variable [

*S*,

*w*]

^{T}so that

*P*[

*S*,

*w*]

^{T}=

*S*. Hence,

*τ*relative to phase relaxation time

_{d}*τ*. This hypothesis is tested and shown in Fig. 2, where the variance of ice supersaturation in an icy cloud is computed using the formulas here presented and its numerical estimation using long time series of 10

_{p}^{6}seconds. The equations are integrated using a simple Euler–Maruyama method with a time step of 10

^{−2}s (Gardiner 2009). It is noted that changing TKE implies a change in the characteristic length scale of the turbulent zone

*L*—here on the order of 1000 m—since the mixing time-scale

*C*

^{−1}is kept constant. Consequently, this figure shows the dependence of ice-supersaturation variance as a function of updraft fluctuation time scales where the homogenization time, namely,

*C*

^{−1}, is the same.

Supersaturation variance as a function of TKE. The blue colors correspond to the variances for the 2D linear system (30) with red-noise updrafts, whereas the black colors correspond to the homogenized 1D Eq. (26), where updrafts become white in time. The dots are obtained by calculating the variance of 10^{6}-s time series with a time step of 10^{−2} s for each value of TKE. The solid lines are the predicted variance using Eqs. (22a) and (35), for the blue and orange curves, respectively. We highlight that when turbulence is less energetic, both estimates become more similar. The ambient conditions in this numerical experiment are *S _{E}* = 0,

*T*= −10°C,

*p*= 50 500 Pa,

*N*= 100 L

_{i}^{−1}.

Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

Supersaturation variance as a function of TKE. The blue colors correspond to the variances for the 2D linear system (30) with red-noise updrafts, whereas the black colors correspond to the homogenized 1D Eq. (26), where updrafts become white in time. The dots are obtained by calculating the variance of 10^{6}-s time series with a time step of 10^{−2} s for each value of TKE. The solid lines are the predicted variance using Eqs. (22a) and (35), for the blue and orange curves, respectively. We highlight that when turbulence is less energetic, both estimates become more similar. The ambient conditions in this numerical experiment are *S _{E}* = 0,

*T*= −10°C,

*p*= 50 500 Pa,

*N*= 100 L

_{i}^{−1}.

Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

Supersaturation variance as a function of TKE. The blue colors correspond to the variances for the 2D linear system (30) with red-noise updrafts, whereas the black colors correspond to the homogenized 1D Eq. (26), where updrafts become white in time. The dots are obtained by calculating the variance of 10^{6}-s time series with a time step of 10^{−2} s for each value of TKE. The solid lines are the predicted variance using Eqs. (22a) and (35), for the blue and orange curves, respectively. We highlight that when turbulence is less energetic, both estimates become more similar. The ambient conditions in this numerical experiment are *S _{E}* = 0,

*T*= −10°C,

*p*= 50 500 Pa,

*N*= 100 L

_{i}^{−1}.

Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

## 4. Diffusional growth and size fluctuations

In the section 2 we presented the full and closed equation for the evolution of supersaturation accounting for the condensational (depositional) growth of droplets (ice crystals) coupled to turbulent updraft fluctuations; see Eq. (6). In section 3, we investigated the properties of the stochastic Squires equation, describing the evolution of supersaturation in the quasi-steady approximation, where condensational growth is neglected. Here, we aim at lifting the quasi-steady assumption, by coupling supersaturation to condensational growth. In this regard, analytical work is limited because of the complexity of the resulting equations (Korolev and Mazin 2003). Nevertheless, approximations have been made to calculate the solutions of the supersaturation equation for small times (Devenish et al. 2016), and to determine the linear growth of variance of droplet radius as time increases (Sardina et al. 2015). In this section we shall provide, first, a brief review of the problem of droplet radius variance increase under random updrafts and, second, present a modified supersaturation equation that accounts for random fluctuations in droplet radius. The novel results are two different probability distributions for supersaturation in the spirit of the previous sections.

*S*≈ 1. Then, by applying the chain rule to Eq. (4), we can write the supersaturation equation coupled to diffusional growth as

*r*

^{2}and makes the analytical work intractable. However, we refer at this stage to appendix C for a note on the analysis of the square root stochastic process.

*rS*is small and close to zero, the mean and variance of

*r*

^{2}evolve according to

*t*≫ 1 and the nonlinear coupling is small (Sardina et al. 2015),

*r*

^{2}= 0 is strictly necessary since it is possible that trajectories of

*r*(

*t*) in Eq. (36b) vanish. When that happens, it means that the particles or droplets in question have evaporated and that the formula for the variance in Eq. (38c) has to be reinitialized once the droplets and particles have reactivated.

While under this framework there is no stationary distribution with finite variance for particle radius, it was shown in Siewert et al. (2017) that if the ambient supersaturation *S _{E}* is negative, i.e., subsaturated, the probability distribution of

*r*

^{2}will possess the structure of an exponential function with a Dirac peak located at

*r*

^{2}= 0, which arises from the boundary condition in Eq. (36c). Indeed, it is expected that the trajectories in the

*S*–

*r*

^{2}plane of Eq. (36) will display cycles, where

*r*

^{2}grows but then vanishes and sticks at the boundary of

*r*

^{2}= 0 for an open interval of time; see Fig. 4 of Siewert et al. (2017).

### a. Fluctuations in droplet radius

In the previous section we clarified that if supersaturation is let to be driven by random turbulent updrafts, the mean-square radius grows linearly in time, and therefore, unbounded Brownian excursions can be expected when solving the condensational growth equation. This implies that variance is not bounded and steady-state distributions of droplet radius have infinite variance. In this section, the target is to compromise between the quasi-steady approximations of section 3 and the coupling with diffusional growth as done in Eq. (36). This is done by accounting for variability in integral radius due to mixing with the cloud’s exterior, perturbations in the particles’ capacitance—particularly for ice particles; see Eq. (4)—and small thermal fluctuations that affect the equilibrium vapor pressure at each droplet’s surface. Small radius fluctuations were already theoretically suggested in the work of Manton (1979) and further explored in Cooper (1989), where the authors consider their effect in the broadening of the droplet size spectrum. Under the quasi-steady approximation, it is concluded that updraft fluctuations alone cannot provoke a broadening of droplet size spectra, but turbulent fluctuations in the integral radius have to be considered; see Eq. (10) in Cooper (1989) and associated comments.

*σ*, which represents the standard deviation of the random fluctuations of the radius variable

_{r}*r*. To simplify the analytical expressions in this section, we shall present the results for

*S*= 0 and

_{E}*S*≈ 1—that together with white-noise fluctuations in droplet size lead to a new equation for supersaturation:

*W*

^{(1)}and

*W*

^{(2)}are two independent Wiener processes. Also, the parameter

*B*in Eq. (15a) has been modified to

*B*=

_{d}*bB*

_{0}

*N*. The first fluctuation term

*N*is factorized out of

*B*. The second fluctuation term

_{d}*f*

_{4}has

*M*momenta, if

*f*

_{4}describes the steady-state statistics when

*σ*tends to zero, which yields the Gaussian distribution of Eq. (28). It is possible, on the other hand, to show the pointwise convergence of Eqs. (41) and (42) to Eq. (28), by means of Laplace’s method; see, e.g., Butler (2007).

_{r}### b. Supersaturation distribution comparison

At this stage we overview the qualitative mathematical features of the calculated supersaturation distributions *f*_{1}, *f*_{2}, and *f*_{3} are those concerning the quasi-steady approximation and they all present distinct characteristics. The positive skew in *f*_{1} predicted by Eq. (23) becomes apparent and highlights the importance of considering the full, nonlinear Squires equation for small values of the nondimensional parameter *α*, as already pointed out in section 3. The time-scale separation between *τ _{d}* and

*τ*is crucial, as is observed in the curves for

_{p}*f*

_{2}and

*f*

_{3}where the latter possesses a relatively more peaked distribution. Upon the introduction of radius fluctuations, two distributions are presented,

*f*

_{4}and

*f*

_{5}. The most characteristic feature of the last two is that they possess thicker tails, as results from the addition of an extra fluctuating term. Indeed, this is more pronounced in Fig. 3b, where a nonexponential tail decay is observed in

*f*

_{4}and

*f*

_{5}. For uncorrelated perturbations in updrafts and particle radius—corresponding to

*f*

_{5}—corresponding

*σ*,

_{r}*f*

_{4}, and

*f*

_{5}would converge to

*f*

_{2}.

Comparison between PDFs. The legend indicates the analytically obtained PDFs *S _{E}* = 0,

*T*= −5°C,

*p*= 50 500 Pa,

^{2}s

^{−2}, and

*N*= 100 L

_{i}^{−1}.

Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

Comparison between PDFs. The legend indicates the analytically obtained PDFs *S _{E}* = 0,

*T*= −5°C,

*p*= 50 500 Pa,

^{2}s

^{−2}, and

*N*= 100 L

_{i}^{−1}.

Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

Comparison between PDFs. The legend indicates the analytically obtained PDFs *S _{E}* = 0,

*T*= −5°C,

*p*= 50 500 Pa,

^{2}s

^{−2}, and

*N*= 100 L

_{i}^{−1}.

Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

## 5. Calculating mixed-phase cloud properties

Mixed-phase clouds constitute a large portion of the global cloud coverage. Altocumulus, altostratus, stratocumulus, and Arctic stratus are instances where mixed-phase conditions have been observed to exist and persist at temperatures as low as −40°C; see, e.g., Korolev et al. (2017). The coexistence of liquid water and ice is, however, thermodynamically unstable so that, in freezing temperatures, ice will inevitably grow at the expense of liquid water (Bergeron 1935). It was suggested in Korolev and Field (2008) a mechanism by which supercooled liquid water can be activated in an initially icy cloud. According to such work, liquid water is activated in an icy cloud if the following two criteria are met: (i) the vertical velocity must exceed a threshold value and (ii) the cloud parcel must be lifted to a threshold altitude. As a consequence, dynamical forcing is responsible for the production of supercooled liquid water and, moreover, different updraft profiles yield different mixed-phase conditions.

It was already advanced in Korolev and Field (2008) that turbulent updrafts could be a mechanism to produced mixed-phase conditions faithful to what is observed, for example, in stratiform clouds; see also Li et al. (2019) for a study on the effects of supersaturation fluctuations on droplet growth on such clouds. Indeed, supersaturation fluctuations can provoke liquid-water saturation conditions in icy parcels, which translates to the activation of supercooled liquid water. In addition, numerical simulations indicate that small-scale turbulent mixing decelerates the ice growth and, thus, extends the lifetime of supercooled water (Hoffmann 2020). In Field et al. (2014), an analytical framework was established for determining mixed-phase properties for icy clouds in turbulent environments. Such framework is based on the analysis of the (ice) supersaturation equation subject to the linear approximation and random turbulent updrafts modeled by white noise; this is revisited in section 3a(1).

*S*to liquid water given a probability distribution for

_{i}*S*. Supersaturation with respect to ice relates to that of water,

_{i}*S*, in terms of the ratio of their respective equilibrium vapor pressures,

_{w}*E*(

_{w}*T*) for water and

*E*(

_{i}*T*) for ice:

*η*(

*T*) =

*E*(

_{w}*T*)/

*E*(

_{i}*T*). As a consequence, supersaturation with respect to ice at water saturation

*S*is given by

_{iw}*S*. According to the present work, such spread is determined by the probability density functions

_{iw}*S*spends above

_{i}*S*or, in other words—by invoking ergodicity (Pavliotis 2014, chapter 1)—the integral of

_{iw}*f*from

_{k}*S*to infinity:

_{iw}*S*larger than

_{i}*S*. As noted in Field et al. (2014), in the Gaussian case the supercooled liquid cloud fraction is given explicitly by

_{iw}*q*⟩

*, for each*

_{k}*k*:

*ρ*is the factor that converts liquid water content to supersaturation values.

_{a}q_{si}The five probability distributions obtained in the previous sections are now used to compute the supercooled liquid cloud fraction and the mean LWC of an adiabatic cloud parcel for a range of free parameters. Such free parameters are the TKE, as a proxy for turbulent forcing, and variance of the droplet radius fluctuations. In Fig. 4 we show the dependence of the mentioned partial moments on values of TKE at two temperatures indicated in the captions: Figs. 4a and 4b show supercooled liquid cloud fraction and mean LWC at −10°C, whereas Figs. 4c and 4d show the same at −5°C. Such statistics where computed using a simple quadrature scheme on the interval [−10, 10] so that all the considered PDFs integrate to unity with a tolerance of 10^{−10}. We observe a monotone dependence on TKE in all the PDFs but for the gamma distribution, which yielded decreasing cloud fractions for values of TKE > 6 m^{2} s^{−2} at −10°C and TKE > 4 m^{2} s^{−2} at −5°C. Such change in trend is due to the displacement of the mode and tail thickness in the gamma distribution as the location factor is altered due to the multiplicative noise. It is noted that the red and orange curves, for *f*_{2} and *f*_{4}, respectively, are almost coinciding.

(a),(c) Supercooled liquid cloud fraction and (b),(d) mean LWC as a function of TKE. The supercooled liquid cloud fraction and mean LWC are calculated using Eqs. (45) and (47), respectively, against TKE for each analytical supersaturation distribution *f*_{2} and *f*_{4}, respectively, are almost coinciding. In (a) and (b), a temperature of −10°C is considered, while in (c) and (d), a temperature of −5°C is considered. The rest of the ambient conditions are *S _{E}* = 0,

*p*= 50 500 Pa,

*N*= 100 L

_{i}^{−1}.

Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

(a),(c) Supercooled liquid cloud fraction and (b),(d) mean LWC as a function of TKE. The supercooled liquid cloud fraction and mean LWC are calculated using Eqs. (45) and (47), respectively, against TKE for each analytical supersaturation distribution *f*_{2} and *f*_{4}, respectively, are almost coinciding. In (a) and (b), a temperature of −10°C is considered, while in (c) and (d), a temperature of −5°C is considered. The rest of the ambient conditions are *S _{E}* = 0,

*p*= 50 500 Pa,

*N*= 100 L

_{i}^{−1}.

Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

(a),(c) Supercooled liquid cloud fraction and (b),(d) mean LWC as a function of TKE. The supercooled liquid cloud fraction and mean LWC are calculated using Eqs. (45) and (47), respectively, against TKE for each analytical supersaturation distribution *f*_{2} and *f*_{4}, respectively, are almost coinciding. In (a) and (b), a temperature of −10°C is considered, while in (c) and (d), a temperature of −5°C is considered. The rest of the ambient conditions are *S _{E}* = 0,

*p*= 50 500 Pa,

*N*= 100 L

_{i}^{−1}.

Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

When droplet size fluctuations are allowed to vary, the first three calculated distributions *f*_{1}, *f*_{2}, and *f*_{3} naturally yield the same statistics for supercooled liquid cloud fraction and mean LWC. Contrarily, when noisy variations in radius are allowed the distributions of *f*_{4} and *f*_{5} are likely to display a dependence on *σ _{r}*. This dependence is shown in Fig. 5, where the supercooled liquid cloud fraction and LWC are calculated as a function

*σ*, for

_{r}*f*

_{2},

*f*

_{4}, and

*f*

_{5}. Figures 5a and 5b show the results at −10°C, whereas Figs. 5c and 5d show the same at −5°C. Because

*f*

_{4}and

*f*

_{5}are expensive to evaluate at small values of

*σ*, ergodic averages are computed instead, i.e., the first equality in Eqs. (45) and (47). Such averages are taken over an integration of Eq. (40) over 10

_{r}^{6}seconds. While

*f*

_{2}is expectedly constant,

*f*

_{4}and

*f*

_{5}start to be dependent for values larger than

^{−8}m

^{2}. On the other hand,

*σ*. We recall that this case corresponds to when the fluctuations in

_{r}*r*are independent of the noisy updraft. Such independence does not hold in LWC, where ⟨

*q*⟩

_{5}appears to grow, as a consequence of the thickening of the tails.

(a),(c) Supercooled liquid cloud fraction and (b),(d) mean LWC as a function of *σ _{r}*. The supercooled liquid cloud fraction and mean LWC are calculated using Eqs. (45) and (47), respectively, against

*σ*for the analytical distributions

_{r}*f*

_{1},

*f*

_{4}, and

*f*

_{5}and for two temperature configurations. For small values of

*σ*, the functions

_{r}*f*

_{4}and

*f*

_{5}are computationally expensive to evaluate so ergodic averages of 10

^{6}seconds are taken instead. In (a) and (b), a temperature of −10°C is considered, while in (b) and (d), a temperature of −5°C is considered. The rest of the ambient conditions are

*S*= 0,

_{E}^{2}s

^{−2},

*p*= 50 500 Pa,

*N*= 100 L

_{i}^{−1}.

Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

(a),(c) Supercooled liquid cloud fraction and (b),(d) mean LWC as a function of *σ _{r}*. The supercooled liquid cloud fraction and mean LWC are calculated using Eqs. (45) and (47), respectively, against

*σ*for the analytical distributions

_{r}*f*

_{1},

*f*

_{4}, and

*f*

_{5}and for two temperature configurations. For small values of

*σ*, the functions

_{r}*f*

_{4}and

*f*

_{5}are computationally expensive to evaluate so ergodic averages of 10

^{6}seconds are taken instead. In (a) and (b), a temperature of −10°C is considered, while in (b) and (d), a temperature of −5°C is considered. The rest of the ambient conditions are

*S*= 0,

_{E}^{2}s

^{−2},

*p*= 50 500 Pa,

*N*= 100 L

_{i}^{−1}.

Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

(a),(c) Supercooled liquid cloud fraction and (b),(d) mean LWC as a function of *σ _{r}*. The supercooled liquid cloud fraction and mean LWC are calculated using Eqs. (45) and (47), respectively, against

*σ*for the analytical distributions

_{r}*f*

_{1},

*f*

_{4}, and

*f*

_{5}and for two temperature configurations. For small values of

*σ*, the functions

_{r}*f*

_{4}and

*f*

_{5}are computationally expensive to evaluate so ergodic averages of 10

^{6}seconds are taken instead. In (a) and (b), a temperature of −10°C is considered, while in (b) and (d), a temperature of −5°C is considered. The rest of the ambient conditions are

*S*= 0,

_{E}^{2}s

^{−2},

*p*= 50 500 Pa,

*N*= 100 L

_{i}^{−1}.

Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

## 6. Discussion

In this paper, the analysis of the stochastic Squires equation has been done, with the aim of providing analytical formulas for the distribution of supersaturation in an adiabatic cloud parcel. Such equation describes the evolution of supersaturation over time by, essentially, taking into account the sources and sinks of relative humidity due to (i) adiabatic cooling via turbulent updrafts and (ii) water vapor condensation onto droplets/particles. Turbulent effects on updrafts and droplet/particle radius are here modeled as a red-noise process, in accordance with the spatial and time correlations of isotropic fluctuations in the inertial regime (Rodean 1996). Thus, we presented a hierarchy of five supersaturation equations that, by using the Fokker–Planck equation, possess analytically tractable distributions whose properties were assessed in the context of mixed phase clouds.

The theoretical results of Field et al. (2014)—here condensed in section 3a(1)—have been here generalized to a wider range of contexts. First of all, the Squires equation is considered in section 3 in its full nonlinear version and has been shown to possess a gamma-like stationary distribution—here denoted as *f*_{1}—that deviates from a Gaussian according to the parameter *α* defined in Eq. (22a). Such parameter, also found in Field et al. (2014), is a nondimensional ratio between the turbulent fluctuation time scales and the phase relaxation coefficient. Thus, as turbulent fluctuations decrease in variance (relative to the microphysical or mixing time scale), *f*_{1} becomes better and better approximated by the Gaussian distribution *f*_{2}. Indeed, this can be seen in section 5 from the computation of partial moments for mixed-phase clouds—mixed cloud fraction and mean LWC—in Fig. 4.

The time-scale separation assumption between updraft fluctuations and supersaturation, needed to compute *f*_{1} and *f*_{2}, is lifted in section 3b. Indeed, *f*_{1} and *f*_{2} are only valid when supersaturation time scales are much greater that the Lagrangian decorrelation *τ _{d}* so that updraft fluctuations become uncorrelated in time; cf. Pavliotis (2014, chapter 11, result 11.1). The resulting distribution

*f*

_{3}is still Gaussian, but possesses a modification in the variance compared to that of

*f*

_{2}; see Eq. (35). In Fig. 2, we show that the variance predicted by

*f*

_{2}and

*f*

_{3}diverge as TKE increases.

The main assumption needed to compute *f*_{1}, *f*_{2}, and *f*_{3} is the quasi-steady approximation, whereby the droplet or ice particle radius is considered constant. When coupling droplet growth and supersaturation in a turbulent environment, it was shown in Sardina et al. (2015) that the long-term variance of droplet squared radius scales linearly in time, similar to Brownian motion. This is revisited here in section 4. We highlight that such result is only valid for short times, since large families of droplets are subject to processes like sedimentation, evaporation, or mixing with exterior dry air that provoke a *memory loss* in collective droplet growth, and hence, the variance formula is reinitialized. Here, we proposed a modification of the supersaturation equation where the sink term—which depends on the integral radius of the droplet/particle population—is allowed to fluctuate randomly, as suggested in earlier work (Manton 1979; Cooper 1989). Two situations are studied: (i) fluctuations in updraft are uncorrelated to those of droplet size and (ii) the source of noise is the same, albeit with different intensities. The resulting model yields analytically tractable probability distributions for supersaturation. The net effect of droplet radius fluctuations is illustrated in Fig. 5, where the supercooled liquid cloud fraction and mean LWC are computed as a function of *σ _{r}*. It is found that correlated radius and updraft fluctuations yield a probability distribution that deviates severely from the quasi-steady approximation by up to +5% in supercooled liquid cloud fraction when

*σ*≥ 10

_{r}^{−6}m. Contrarily, the mean LWC is negatively correlated with fluctuations in

*σ*in case of

_{r}*f*

_{4}. Regarding the uncorrelated sources of noise, the supercooled liquid cloud fraction appears to be weakly dependent in

*σ*. On the other hand, mean LWC correlates positively is droplet size fluctuation variance. All the obtained PDFs are also qualitatively compared in Fig. 3.

_{r}There is still a need to provide a more quantitative verification of the formulas here presented. However, the formulas and discussions in this paper suggest that the widely used linear approximation for supersaturation evolution—see Eq. (26)—is limited to clouds experiencing low supersaturation, where the quasi-steady approximation is valid and updrafts decorrelate instantly with respect to phase relaxation time scales. The mixed-phase cloud stochastic parameterization scheme developed in Furtado et al. (2016) is based on these assumptions, and hence, future work should be oriented toward discerning which specific atmospheric conditions are appropriate for each of the here calculated supersaturation PDFs

On a more theoretical note, a deeper investigation of Eq. (6) would be of great interest in this stochastic framework. One step forward would be to impose a characteristic time for the loss of memory, so that the integrodifferential equation can be replaced by a simpler expression, possibly some form of noise with suitable time-decorrelation properties. We anticipate that this would entail technical difficulties due to the square root nonlinearity—here minimally tackled in appendix C—so research should be oriented in this direction. In general, we belief that this statistical-physics approach can be extended to more general contexts, possibly, by including more microphysical processes that affect the growth of liquid droplets or ice particles and, hence, the overall regulation of a cloud’s supersaturation budget.

## Acknowledgments.

The authors thank S. Roncoroni, P. Field, and B. Devenish for their comments, suggestions, and kind reception at the Met Office. MSG is grateful to the Mathematics of Planet Earth Centre for Doctoral Training (MPE CDT) for making this collaboration possible. MSG acknowledges and is grateful for the support of the Institute of Mathematics and its Applications (Grant Number SGS21/08). MSG is thankful to I. Koren, M. D. Chekroun, the cloud physics group, and the graduate school at the Weizmann Institute of Science for providing a most inspiring environment.

## Data availability statement.

No data have been used in this publication apart from the numerical integration of the equations here presented. The numerical schemes employed are found in https://doi.org/10.5281/zenodo.7904642.

## APPENDIX A

### List of Some Used Notations and Symbols

Table A1 provides a list of symbols that appear in the main text.

List of symbols.

## APPENDIX B

### Stationary Supersaturation Distribution

*S*and assuming that the stationary distribution

*f*and

*S*tends to infinity,

*f*(

*S*) goes to zero asymptotically. We now calculate the normalization constant

*C*, for which we employ the parameter

*α*like in Eq. (22a):

*f*/

*C*gives Eq. (21). This process is repeated for every probability distribution

#### The general case

*W*is a standard Wiener process and

_{t}*V*and

*σ*≠ 0 have the regularity so that solutions distribute according to a smooth probability density function (Gardiner 2009; Pavliotis 2014). The associated Fokker–Planck equation is

*f*is a probability density and a function of

*x*and

*t*. The stationary distribution of Eq. (B7) is obtained by setting

*∂*= 0, and solving for

_{t}f*f*:

*f*is assumed to vanish, to

*x*and rearranging to make the equation homogeneous,

## APPENDIX C

### The Square Root Process

*δ*(

_{S}*t*), decorrelate instantly and have zero mean, such expression is rewritten as

*A*

_{∘}. Deriving the statistics of a squared root stochastic process is difficult, although in the limit of variance fluctuations of

*r*

^{2}begin small, or when the diffusional constant

*r*(0)

^{2}. This approach assumes that

*r*(0) is the mean radius and that fluctuations

*δ*(

_{S}*t*) are so small that

*r*

^{2}remains positive. This result is general and can be applied to any square root random variable, with a positive mean, and in the limit of small variance.

To support this analytical expansion, we numerically sampled an adimensional random variable *X*, normally distributed with mean 2 and standard deviation *σ _{X}*, where the latter takes 250 equispaced values between 10

^{−4}and 0.5. The sample is of size 10

^{5}draws. With this set of data, we are able to numerically estimate the mean and variance of the square root random variable,

*σ*. The truncation is done at

_{X}*σ*becomes larger, a moderate deviation is observed, as expected.

_{X}Variance and mean of the square root random variable. The black dots are calculated as follows: for each value of *σ _{X}*, the variance and mean of the random variable

^{5}draws of a normal random variable

*X*with mean 2 and standard deviation

*σ*. The blue and orange solid lines indicate the truncated predictions of Eq. (C2) for the variance and mean, respectively.

_{X}Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

Variance and mean of the square root random variable. The black dots are calculated as follows: for each value of *σ _{X}*, the variance and mean of the random variable

^{5}draws of a normal random variable

*X*with mean 2 and standard deviation

*σ*. The blue and orange solid lines indicate the truncated predictions of Eq. (C2) for the variance and mean, respectively.

_{X}Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

Variance and mean of the square root random variable. The black dots are calculated as follows: for each value of *σ _{X}*, the variance and mean of the random variable

^{5}draws of a normal random variable

*X*with mean 2 and standard deviation

*σ*. The blue and orange solid lines indicate the truncated predictions of Eq. (C2) for the variance and mean, respectively.

_{X}Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

## REFERENCES

Abade, G. C., W. W. Grabowski, and H. Pawlowska, 2018: Broadening of cloud droplet spectra through eddy hopping: Turbulent entraining parcel simulations.

,*J. Atmos. Sci.***75**, 3365–3379, https://doi.org/10.1175/JAS-D-18-0078.1.Baker, M. B., R. E. Breidenthal, T. W. Choularton, and J. Latham, 1984: The effects of turbulent mixing in clouds.

,*J. Atmos. Sci.***41**, 299–304, https://doi.org/10.1175/1520-0469(1984)041<0299:TEOTMI>2.0.CO;2.Bartlett, J. T., and P. R. Jonas, 1972: On the dispersion of the sizes of droplets growing by condensation in turbulent clouds.

,*Quart. J. Roy. Meteor. Soc.***98**, 150–164, https://doi.org/10.1002/qj.49709841512.Bergeron, T., 1935: On the physics of clouds and precipitation.

*Proces Verbaux de l’Association de Meteorologie*, Lisbon, Portugal, International Union of Geodesy and Geophysics, 156–178.Butler, R. W., 2007: Exponential families and tilted distributions.

*Saddlepoint Approximations with Applications*, Cambridge Series in Statistical and Probabilistic Mathematics, Vol. 22, Cambridge University Press, 145–182, https://doi.org/10.1017/CBO9780511619083.006.Chen, S., and Coauthors, 2023: Mixed-phase direct numerical simulation: Ice growth in cloud-top generating cells.

,*Atmos. Chem. Phys.***23**, 5217–5231, https://doi.org/10.5194/acp-23-5217-2023.Cooper, W. A., 1989: Effects of variable droplet growth histories on droplet size distributions. Part I: Theory.

,*J. Atmos. Sci.***46**, 1301–1311, https://doi.org/10.1175/1520-0469(1989)046<1301:EOVDGH>2.0.CO;2.Devenish, B. J., K. Furtado, and D. J. Thomson, 2016: Analytical solutions of the supersaturation equation for a warm cloud.

,*J. Atmos. Sci.***73**, 3453–3465, https://doi.org/10.1175/JAS-D-15-0281.1.Einstein, A., 1905: Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen.

,*Ann. Phys.***322**, 549–560, https://doi.org/10.1002/andp.19053220806.Eytan, E., A. Khain, M. Pinsky, O. Altaratz, J. Shpund, and I. Koren, 2022: Shallow cumulus properties as captured by adiabatic fraction in high-resolution LES simulations.

,*J. Atmos. Sci.***79**, 409–428, https://doi.org/10.1175/JAS-D-21-0201.1.Field, P. R., A. A. Hill, K. Furtado, and A. Korolev, 2014: Mixed-phase clouds in a turbulent environment. Part 2: Analytic treatment.

,*Quart. J. Roy. Meteor. Soc.***140**, 870–880, https://doi.org/10.1002/qj.2175.Furtado, K., P. R. Field, I. A. Boutle, C. J. Morcrette, and J. M. Wilkinson, 2016: A physically based subgrid parameterization for the production and maintenance of mixed-phase clouds in a general circulation model.

,*J. Atmos. Sci.***73**, 279–291, https://doi.org/10.1175/JAS-D-15-0021.1.Gardiner, C., 2009:

*Stochastic Methods: A Handbook for the Natural and Social Sciences*. Springer-Verlag, 447 pp.Grabowski, W. W., and G. C. Abade, 2017: Broadening of cloud droplet spectra through eddy hopping: Turbulent adiabatic parcel simulations.

,*J. Atmos. Sci.***74**, 1485–1493, https://doi.org/10.1175/JAS-D-17-0043.1.Hoffmann, F., 2020: Effects of entrainment and mixing on the Wegener–Bergeron–Findeisen process.

,*J. Atmos. Sci.***77**, 2279–2296, https://doi.org/10.1175/JAS-D-19-0289.1.Khain, A. P., and M. Pinsky, 2018:

*Physical Processes in Clouds and Cloud Modeling*. Cambridge University Press, 640 pp., https://doi.org/10.1017/9781139049481.Korolev, A. V., and I. P. Mazin, 2003: Supersaturation of water vapor in clouds.

,*J. Atmos. Sci.***60**, 2957–2974, https://doi.org/10.1175/1520-0469(2003)060<2957:SOWVIC>2.0.CO;2.Korolev, A. V., and P. R. Field, 2008: The effect of dynamics on mixed-phase clouds: Theoretical considerations.

,*J. Atmos. Sci.***65**, 66–86, https://doi.org/10.1175/2007JAS2355.1.Korolev, A. V., and Coauthors, 2017: Mixed-phase clouds: Progress and challenges.

*Ice Formation and Evolution in Clouds and Precipitation: Measurement and Modeling Challenges*,*Meteor. Monogr.*, No. 58, Amer. Meteor. Soc., https://doi.org/10.1175/AMSMONOGRAPHS-D-17-0001.1.Li, X.-Y., G. Svensson, A. Brandenburg, and N. E. L. Haugen, 2019: Cloud-droplet growth due to supersaturation fluctuations in stratiform clouds.

,*Atmos. Chem. Phys.***19**, 639–648, https://doi.org/10.5194/acp-19-639-2019.Manton, M. J., 1979: On the broadening of a droplet distribution by turbulence near cloud base.

,*Quart. J. Roy. Meteor. Soc.***105**, 899–914, https://doi.org/10.1002/qj.49710544613.McGraw, R., and Y. Liu, 2006: Brownian drift-diffusion model for evolution of droplet size distributions in turbulent clouds.

,*Geophys. Res. Lett.***33**, L03802, https://doi.org/10.1029/2005GL023545.Paoli, R., and K. Shariff, 2009: Turbulent condensation of droplets: Direct simulation and a stochastic model.

,*J. Atmos. Sci.***66**, 723–740, https://doi.org/10.1175/2008JAS2734.1.Pavliotis, G. A., 2014:

*Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations*. Vol. 60, Springer, 339 pp.Pavliotis, G. A., and A. M. Stuart, 2008:

*Multiscale Methods: Averaging and Homogenization*. Springer, 310 pp.Prabha, T. V., A. Khain, R. S. Maheshkumar, G. Pandithurai, J. R. Kulkarni, M. Konwar, and B. N. Goswami, 2011: Microphysics of premonsoon and monsoon clouds as seen from in situ measurements during the Cloud Aerosol Interaction and Precipitation Enhancement Experiment (CAIPEEX).

,*J. Atmos. Sci.***68**, 1882–1901, https://doi.org/10.1175/2011JAS3707.1.Prabhakaran, P., A. S. M. Shawon, G. Kinney, S. Thomas, W. Cantrell, and R. A. Shaw, 2020: The role of turbulent fluctuations in aerosol activation and cloud formation.

,*Proc. Natl. Acad. Sci. USA***117**, 16 831–16 838, https://doi.org/10.1073/pnas.2006426117.Pruppacher, H. K., and J. D. Klett, 2010:

*Microphysics of Clouds and Precipitation*. Springer, 954 pp.Risken, H., 1989:

*The Fokker-Planck Equation: Methods of Solution and Applications*. 2nd ed. Springer-Verlag, 472 pp.Rodean, H. C., 1996:

*Stochastic Lagrangian Models of Turbulent Diffusion. Meteor. Monogr*., No. 48, Amer. Meteor. Soc., 84 pp.Rogers, R. R., and M. K. Yau, 1989:

*A Short Course in Cloud Physics*. 3rd ed. Pergamon Press, 304 pp.Saito, I., T. Gotoh, and T. Watanabe, 2019: Broadening of cloud droplet size distributions by condensation in turbulence.

,*J. Meteor. Soc. Japan***97**, 867–891, https://doi.org/10.2151/jmsj.2019-049.Sardina, G., F. Picano, L. Brandt, and R. Caballero, 2015: Continuous growth of droplet size variance due to condensation in turbulent clouds.

,*Phys. Rev. Lett.***115**, 184501, https://doi.org/10.1103/PhysRevLett.115.184501.Shaw, R. A., 2003: Particle-turbulence interactions in atmospheric clouds.

,*Annu. Rev. Fluid Mech.***35**, 183–227, https://doi.org/10.1146/annurev.fluid.35.101101.161125.Siebert, H., and R. A. Shaw, 2017: Supersaturation fluctuations during the early stage of cumulus formation.

,*J. Atmos. Sci.***74**, 975–988, https://doi.org/10.1175/JAS-D-16-0115.1.Siewert, C., J. Bec, and G. Krstulovic, 2017: Statistical steady state in turbulent droplet condensation.

,*J. Fluid Mech.***810**, 254–280, https://doi.org/10.1017/jfm.2016.712.Squires, P., 1952: The growth of cloud drops by condensation.

,*Aust. J. Sci. Res.***5**, 59.Uhlenbeck, G. E., and L. S. Ornstein, 1930: On the theory of the Brownian motion.

,*Phys. Rev.***36**, 823–841, https://doi.org/10.1103/PhysRev.36.823.Zelinka, M. D., T. A. Myers, D. T. McCoy, S. Po-Chedley, P. M. Caldwell, P. Ceppi, S. A. Klein, and K. E. Taylor, 2020: Causes of higher climate sensitivity in CMIP6 models.

,*Geophys. Res. Lett.***47**, e2019GL085782, https://doi.org/10.1029/2019GL085782.